URGENT: Differential calculus

Okay, my english is not perfect, but I hope you can understand what I'm talking about.

I'm having this subject about movement and air resistance. We're watching an object dropping in an area with air resistance. The resulting force on the object is (using Newton's 2. law):

F_res = -F_grav + F_air <=> ma = -mg + k*v^2

Let's introduce s(t) (don't know what it's called in english), and thereby s'(t) = v(t) and s''(t) = a(t).

After some paraphrasing, we get a(t) = k/m * v^2 - g <=> s''(t) = C * (s'(t))^2 - g with C = k/m

To find v(t) and s(t) we integrate, so

v(t) = 1/3 * C * t * (s(t))^3 - g * t + v_0 and

s(t) = 1/12 * C * t^2 * ((int.)s(t)dt)^4 - g * t^2 + v_0*t + s_0

Is this correct? How do I find s(t)??? My calculator can't find it.. Maybe I'm typing something wrong or I've done something wrong.. It's really urgent, please help!